1

Bias correction in data assimilation

Hans Hersbach and Dick Dee

ECMWF

Meteorological Training Course Data Assimilation

12-13 March 2014

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Overview of this lecture

In this lecture the variational bias correction scheme (VarBC) as used at ECMWF is explained. VarBC replaced the tedious job of estimating observation bias off-line for each satellite instrument or in-situ network by an automatic self-adaptive system. This is achieved by making the bias estimation an integral part of the ECMWF variational data assimilation system, where now both the initial model state and observation bias estimates are updated simultaneously. By the end of the session you should be able to realize that: 1. many observations are biased, and that the characteristics of bias varies widely between

types of instruments, 2. separation between model bias and observation bias is often difficult, 3. the success of an adaptive system implicitly relies on a redundancy in the underlying

observing system.

[ ] [ ]h(x)yRh(x)yx)(xBx)(xJ(x) 1Tb1Tb −−+−−= −−

background constraint observational constraint

Bias: Ignorance is bliss…

So… where is the bias term in this equation?

Everyone knows that models are biased

Not everyone knows that most observations are biased as well

Outline

• Introduction – Biases in models, observations, and observation operators – Implications for data assimilation

• Variational analysis and correction of observation bias

– The need for an adaptive system – Variational bias correction (VarBC)

• Extension to other types of observations

• Limitations due to the effects of model bias

Model bias: Systematic Day-3 Z500 errors in three different forecast models

ECMWF Meteo-France DWD

Different models often have similar error characteristics Period DJF 2001-2003

Model bias: Seasonal variation in upper-stratospheric model errors

40hPa (22km)

0.1hPa (65km)

T255L60 model currently used for the ERA-Interim reanalysis

Summer: Radiation, ozone?

Winter: Gravity-wave drag?

Observation bias: Radiosonde temperature observations

observed – ERA-40 background at Saigon (200 hPa, 0 UTC)

Bias changes due to change of equipment

Daytime warm bias due to radiative heating of the temperature sensor (depends on solar elevation and equipment type)

Mean temperature anomalies for different solar elevations

Observation and observation operator bias: Satellite radiances

Diurnal bias variation in a geostationary satellite

Air-mass dependent bias (AMSU-A channel 14)

Constant bias (HIRS channel 5)

Monitoring the background departures (averaged in time and/or space):

nadir high zenith angle

Bias depending on scan position (AMSU-A ch 7) NH Trop SH

high zenith angle

Obs

-FG

bia

s [K

] O

bs-F

G b

ias

[K]

Observation and observation operator bias: Satellite radiances

Monitoring the background departures (averaged in time and/or space):

Obs

-FG

bia

s [K

] O

bs-F

G b

ias

[K]

HIRS channel 5 (peaking around 600hPa) on NOAA-14 satellite has +2.0K radiance bias against FG.

Same channel on NOAA-16 satellite has no radiance bias against FG.

NOAA-14 channel 5 has an instrument bias.

Observation and observation operator bias: Satellite radiances

METEOSAT-9, 13.4µm channel:

Drift in bias due to ice-build up on sensor:

Sensor decontamination Obs

–FG

Bia

s

Different bias for HIRS due to change in spectroscopy used in the radiative transfer model:

Obs-FG bias [K]

Channel num

ber

Old spectroscopy

New spectroscopy

Other common causes for biases in radiative transfer:

• Bias in assumed concentrations of atmospheric gases (e.g., CO2, aerosols)

• Neglected effects (e.g., clouds) • Incorrect spectral response function • ….

Implications for data assimilation: Bias problems in a nutshell

• Observations and observation operators have biases, which may change over time

– Daytime warm bias in radiosonde measurements of stratospheric temperature; radiosonde equipment changes

– Biases in cloud-drift wind data due to problems in height assignment – Biases in satellite radiance measurements and radiative transfer models

• Models have biases, and changes in observational coverage over time may change the

extent to which observations correct these biases

– Stratospheric temperature bias modulated by radiance assimilation – This is especially important for reanalysis (trend analysis)

• Data assimilation methods are primarily designed to correct small (random) errors in

the model background

– Large corrections generally introduce spurious signals in the assimilation – Likewise, inconsistencies among different parts of the observing system lead to

all kinds of problems

Implications for data assimilation: The effect of model bias on trend estimates

Most assimilation systems assume unbiased models and unbiased data

Biases in models and/or data can induce spurious trends in the assimilation

Based on monthly CRUTEM2v data (Jones and Moberg, 2003)

Based on ERA-40 reanalysis

Implications for data assimilation: ERA-40 surface temperatures compared to land-station values

Northern hemisphere

Based on ERA-40 model simulation (with SST/sea-ice data)

Surface air temperature anomaly (oC) with respect to 1987-2001

Outline

• Introduction – Biases in models, observations, and observation operators – Implications for data assimilation

• Variational analysis and correction of observation bias

– The need for an adaptive system – The variational bias correction scheme: VarBC

• Extension to other types of observations

• Limitations due to the effects of model bias

Variational analysis and bias correction: A brief review of variational data assimilation

[ ] [ ]h(x)yRh(x)yx)(xBx)(xJ(x) 1Tb1Tb −−+−−= −−Minimise

background constraint (Jb) observational constraint (Jo)

• The input xb represents past information propagated by the forecast model (the model background) • The input [y – h(xb)] represents the new information entering the system (the background departures) • The function h(x) represents a model for simulating observations (the observation operator) • Minimising the cost function J(x) produces an adjustment to the model background

based on all used observations (the analysis)

Variational analysis and bias correction: Error sources in the input data

[ ] [ ]h(x)yRh(x)yx)(xBx)(xJ(x) 1Tb1Tb −−+−−= −−Minimise

background constraint (Jb) observational constraint (Jo)

• Errors in the input [y – h(xb)] arise from:

• errors in the actual observations • errors in the model background • errors in the observation operator

• There is no general method for separating these different error sources

• we only have information about differences • there is no true reference in the real world!

• The analysis does not respond well to conflicting input information A lot of work is done to remove biases prior to assimilation:

• ideally by removing the cause • in practise by careful comparison against other data

The need for an adequate bias model

Diurnal bias variation in a geostationary satellite Constant bias (HIRS channel 5)

Prerequisite for any bias correction is a good model for the bias (b(x,β)): • Ideally, guided by the physical origins of the bias. • In practice, bias models are derived empirically from observation

monitoring.

nadir high zenith angle

Bias depending on scan position (AMSU-A ch 7)

high zenith angle

1.7

1.0

0.0

-1.0

Air-mass dependent bias (AMSU-A ch 10)

Obs

-FG

bia

s [K

] O

bs-F

G b

ias

[K]

Scan bias and air-mass dependent bias for each satellite/sensor/channel were estimated off-line from background departures, and stored in files (Harris and Kelly 2001)

Satellite radiance bias correction at ECMWF, prior to 2006

errorn observatio random

positionscan latitude,

)(

)(

10

=

+=

=

∑ =obs

iN

i iair

scanscan

e

xpb

bb

ββ

obsairscan exbbxhy +++= )()(Error model for brightness temperature data:

where

Periodically estimate scan bias and predictor coefficients: • typically 2 weeks of background departures • 2-step regression procedure • careful masking and data selection

Average the background departures:

Predictors, for instance: • 1000-300 hPa thickness • 200-50 hPa thickness • surface skin temperature • total precipitable water

The need for an adaptive bias correction system

• The observing system is increasingly complex and constantly changing • It is dominated by satellite radiance data:

• biases are flow-dependent, and may change with time • they are different for different sensors • they are different for different channels

• How can we manage the bias corrections for all these different components? • This requires a consistent approach and a flexible, automated system

The bias in a given instrument/channel (bias group) is described by (a few) bias parameters: typically, these are functions of air-mass and scan-position (the predictors) These parameters can be estimated in a variational analysis along with the model state (Derber and Wu, 1998 at NCEP, USA)

The Variational bias correction scheme: The general idea

The standard variational analysis minimizes

Modify the observation operator to account for bias:

Include the bias parameters in the control vector:

Minimize instead

h(x)][yRh(x)][yx)(xBx)(xJ(x) TbxT

b −−+−−=−− 11

What is needed to implement this:

1. The modified operator and its TL + adjoint 2. A cycling scheme for updating the bias parameter estimates 3. An effective preconditioner for the joint minimization problem

Variational bias correction: The modified analysis problem

Jb: background constraint

Jo: observation constraint

[ ] [ ]h(x)yRh(x)yx)(xBx)(x(x) 1Tb1Tb −−+−−= −−J

The original problem:

[ ] [ ]h(x)β)(x,byRh(x)β)(x,byβ)(βBβ)(βx)(xBx)(xβ)J(x,

o1T

o

b1

βT

bb1

xT

b

−−−−+

−−+−−=−

−−

Jb: background constraint for x Jβ: background constraint for β

Jo: bias-corrected observation constraint

The modified problem:

Parameter estimates from previous analysis

Example 1: Spinning up new instruments – IASI on MetOp A

• IASI is a high-resolution interferometer with 8461 channels • Initially unstable – data gaps, preprocessing changes

Variational bias correction smoothly handles the abrupt change in bias:

• initially QC rejects most data from this channel • the variational analysis adjusts the bias estimates • bias-corrected data are gradually allowed back in

• no shock to the system!

Example 2: NOAA-9 MSU channel 3 bias corrections (cosmic storm)

200 hPa temperature departures from radiosonde observations

Example 3: Fit to conventional data

Introduction of VarBC in ECMWF operations

Outline

• Introduction – Biases in models, observations, and observation operators – Implications for data assimilation

• Variational analysis and correction of observation bias

– The need for an adaptive system – Variational bias correction (VarBC)

• Extension to other types of observations

• Limitations due to the effects of model bias

Extension to other types of observations

Current bias ‘classes’ in the ECMWF operational system: • Radiances: clear sky/all sky, infrared/microwave, polar/geostationary • Total column ozone: currently only OMI • Aircraft data: one group per aircraft • Total column water vapour: ENVISAT MERIS until April 2012 • Ground-based radar precipitation: one group embracing US stations Other automated bias corrections, but outside 4D-Var: • Surface pressure • Radiosonde temperature and humidity Specific: • ERA-Interim: VarBC for radiances only • ERA-20C: the 20th century reanalysis using surface observations only • MACC: atmospheric composition

VarBC for satellite radiances

• ~1,500 channels (~40 sensors on ~25 different satellites) • Anchored to each other, GPS-RO, and all conventional observations • Bias model: β0 + ∑βi pi(model state) + ∑βj pj(instrument state) (~11,400 parameters in total)

VarBC for ozone

• OMI, (SCIAMACHY, GOMOS, SEVIRI, GOME2, GOME in past) • Anchored to SBUV/2 • Bias model: β0 + β1 x solar elevation

VarBC for aircraft temperature

• For each aircraft separately (~5000 distinct aircraft) • Anchored to all temperature-sensitive observations • Bias model: β0 + β1 x ascent rate + β2 x descent rate

Average temperature departures for the northern hemisphere during a 2-week period

Aircraft Radiosonde Control Impact

Outline

• Introduction – Biases in models, observations, and observation operators – Implications for data assimilation

• Variational analysis and correction of observation bias

– The need for an adaptive system – Variational bias correction (VarBC)

• Extension to other types of observations

• Limitations due to the effects of model bias

Limitations of VarBC: Interaction with model bias

[ ] [ ]h(x)β)b(x,yRh(x)β)b(x,yβ)(βBβ)(βx)(xBx)(xβ)J(x,

1T

b1

βT

bb1

xT

b

−−−−+

−−+−−=−

−−

VarBC introduces extra degrees of freedom in the variational analysis, to help improve the fit to the (bias-corrected) observations:

It does not work as well when there are large model biases and few observations to constrain them:

model

observations

VarBC is not designed to correct model biases: Need for a weak-constraints 4D-Var (Trémolet)

It works well (even if the model is biased) when the analysis is strongly constrained by observations:

model

abundant observations

Summary

Biases are everywhere: • Most observations cannot be usefully assimilated without bias adjustments • Off-line bias tuning for satellite data is practically impossible • Bias parameters can be estimated and adjusted during the assimilation, using all available information • Variational bias correction works best in situations where:

• there is sufficient redundancy in the data; or • there are no large model biases

Challenges: • How to develop good bias models for observations • How to separate observation bias from model bias

Additional information

Harris and Kelly, 2001: A satellite radiance-bias correction scheme for data assimilation. Q. J. R. Meteorol. Soc., 127, 1453-1468

Derber and Wu, 1998: The use of TOVS cloud-cleared radiances in the NCEP SSI analysis system. Mon. Wea. Rev., 126, 2287-2299

Dee, 2004: Variational bias correction of radiance data in the ECMWF system. Pp. 97-112 in Proceedings of the ECMWF workshop on assimilation of high spectral resolution sounders in NWP, 28 June-1 July 2004, Reading, UK

Dee, 2005: Bias and data assimilation. Q. J. R. Meteorol. Soc., 131, 3323-3343 Dee and Uppala, 2009: Variational bias correction of satellite radiance data in the

ERA-Interim reanalysis. Q. J. R. Meteorol. Soc., 135, 1830-1841 Feel free to contact me with questions: Hans.Hersbach@ecmwf.int

VarBC for total column water vapour

• ENVISAT/MERIS until April 2012 • Anchored to all other humidity-sensitive observations • Bias model: β0 + β1 x TCWV(model state)

Bias correction in data assimilationOverview of this lectureBias: Ignorance is bliss…OutlineModel bias:�Systematic Day-3 Z500 errors in three different forecast modelsModel bias: �Seasonal variation in upper-stratospheric model errorsObservation bias:�Radiosonde temperature observationsObservation and observation operator bias: �Satellite radiancesObservation and observation operator bias: �Satellite radiancesObservation and observation operator bias: �Satellite radiancesImplications for data assimilation:�Bias problems in a nutshellImplications for data assimilation:�The effect of model bias on trend estimatesImplications for data assimilation: �ERA-40 surface temperatures compared to land-station valuesOutlineVariational analysis and bias correction:�A brief review of variational data assimilationVariational analysis and bias correction:�Error sources in the input dataThe need for an adequate bias modelSatellite radiance bias correction at ECMWF, prior to 2006The need for an adaptive bias correction systemThe Variational bias correction scheme:�The general ideaVariational bias correction: �The modified analysis problemExample 1: �Spinning up new instruments – IASI on MetOp AExample 2:�NOAA-9 MSU channel 3 bias corrections (cosmic storm)Example 3:�Fit to conventional dataOutlineExtension to other types of observationsVarBC for satellite radiancesVarBC for ozoneVarBC for aircraft temperatureOutlineLimitations of VarBC:�Interaction with model biasSummaryAdditional informationVarBC for total column water vapour